K-12 Quadratic Formula Guide

📐 Complete Quadratic Formula Guide 📐

Everything You Need to Know About Quadratic Equations (K-12)

🌟 The Quadratic Formula 🌟

x = (-b ± √(b² - 4ac)) / 2a

Used to solve: ax² + bx + c = 0

Standard Form of Quadratic Equation

ax² + bx + c = 0

Where:
• a = coefficient of x² (must not be 0)
• b = coefficient of x
• c = constant term
• x = variable (what we're solving for)

Different Forms of Quadratic Equations

Standard Form

ax² + bx + c = 0

Example: 2x² + 5x - 3 = 0

Vertex Form

y = a(x - h)² + k

Vertex at (h, k)
Example: y = 2(x - 3)² + 1

Factored Form

y = a(x - r₁)(x - r₂)

Roots at r₁ and r₂
Example: y = 2(x - 1)(x + 3)

Intercept Form

y = a(x - p)(x - q)

x-intercepts at p and q
Example: y = (x - 2)(x - 5)

The Discriminant (b² - 4ac)

Δ = b² - 4ac

The discriminant tells us how many and what type of solutions the quadratic equation has!

Discriminant Value	Number of Solutions	Type of Solutions
Δ > 0 (positive)	2 distinct solutions	Two different real numbers
Δ = 0	1 solution (repeated)	One real number (double root)
Δ < 0 (negative)	No real solutions	Two complex/imaginary numbers

All Methods to Solve Quadratic Equations

Method 1: Quadratic Formula

1 Write equation in standard form: ax² + bx + c = 0

2 Identify values of a, b, and c

3 Substitute into formula: x = (-b ± √(b² - 4ac)) / 2a

4 Calculate discriminant: b² - 4ac

5 Solve for x using both + and - in ±

Method 2: Factoring

Best when the equation can be written as a product of two binomials

Common Factoring Patterns

Difference of Squares

a² - b² = (a + b)(a - b)

Example: x² - 9 = (x + 3)(x - 3)

Perfect Square Trinomial

a² + 2ab + b² = (a + b)²

a² - 2ab + b² = (a - b)²

Example: x² + 6x + 9 = (x + 3)²

General Trinomial

x² + bx + c = (x + m)(x + n)

where m + n = b and m × n = c

AC Method

ax² + bx + c

Find two numbers that multiply to ac and add to b

Steps for Factoring

1 Write in standard form and factor out GCF if possible

2 Look for special patterns (difference of squares, perfect square)

3 If trinomial, find two numbers that multiply to ac and add to b

4 Write as product of two binomials

5 Use Zero Product Property: Set each factor = 0 and solve

Method 3: Completing the Square

x² + bx + (b/2)² = (x + b/2)²

Steps for Completing the Square

1 Move constant c to the right side: ax² + bx = -c

2 Divide everything by a (if a ≠ 1): x² + (b/a)x = -c/a

3 Take half of the x coefficient, square it: (b/2a)²

4 Add this to both sides of equation

5 Factor left side as perfect square: (x + b/2a)²

6 Take square root of both sides (± on right)

7 Solve for x

Method 4: Graphing

The solutions are where the parabola crosses the x-axis (x-intercepts)

Key Features of Quadratic Graphs

Vertex

x = -b / 2a

y = f(-b/2a)

The highest or lowest point

Axis of Symmetry

x = -b / 2a

Vertical line through vertex

Y-intercept

y = c

Point where graph crosses y-axis: (0, c)

Direction

If a > 0: Opens up (∪)

If a < 0: Opens down (∩)

Important Properties & Relationships

Sum and Product of Roots

Sum of Roots

r₁ + r₂ = -b/a

Product of Roots

r₁ × r₂ = c/a

Converting Between Forms

Standard to Vertex Form:
y = ax² + bx + c → y = a(x - h)² + k
where h = -b/2a and k = c - b²/4a

Standard to Factored Form:
y = ax² + bx + c → y = a(x - r₁)(x - r₂)
where r₁ and r₂ are the roots (solutions)

Worked Examples

Example 1: Using the Quadratic Formula

Solve: 2x² + 7x - 4 = 0

a = 2, b = 7, c = -4

Discriminant: b² - 4ac = 7² - 4(2)(-4) = 49 + 32 = 81

x = (-7 ± √81) / (2·2) = (-7 ± 9) / 4

x₁ = (-7 + 9) / 4 = 2/4 = 1/2

x₂ = (-7 - 9) / 4 = -16/4 = -4

Solutions: x = 1/2 and x = -4

Example 2: Factoring

Solve: x² + 5x + 6 = 0

Find two numbers that multiply to 6 and add to 5

Numbers: 2 and 3 (2 × 3 = 6, 2 + 3 = 5)

(x + 2)(x + 3) = 0

x + 2 = 0 → x = -2

x + 3 = 0 → x = -3

Solutions: x = -2 and x = -3

Example 3: Completing the Square

Solve: x² + 6x + 2 = 0

x² + 6x = -2

Take half of 6: 6/2 = 3, square it: 3² = 9

x² + 6x + 9 = -2 + 9

(x + 3)² = 7

x + 3 = ±√7

x = -3 ± √7

Solutions: x = -3 + √7 and x = -3 - √7

Example 4: Perfect Square

Solve: x² - 10x + 25 = 0

Recognize as perfect square: (x - 5)²

(x - 5)² = 0

x - 5 = 0

x = 5

Solution: x = 5 (double root, Δ = 0)

🧮 Interactive Quadratic Formula Calculator

Enter the coefficients for ax² + bx + c = 0

Coefficient a (x² term):

Coefficient b (x term):

Coefficient c (constant):

📋 Quick Reference Chart

Method	Best Used When	Difficulty
Factoring	Equation factors easily, integer solutions	Easy-Medium
Quadratic Formula	Always works! Use when factoring is difficult	Medium
Completing the Square	Deriving quadratic formula, vertex form	Medium-Hard
Square Root Method	No bx term (b = 0), like x² = 16	Easy
Graphing	Visual understanding, approximate solutions	Easy-Medium